abacus attention 7-20-16 - stanford language and...

Abacus and Visual Attention 1

The role of design and training in artifact expertise: The case of mental abacus and visual

attention

Mahesh Srinivasan1, Katie Wagner2, Michael C. Frank3, & David Barner2

1University of California, Berkeley

2University of California, San Diego

3Stanford University

Contact Information:

Mahesh Srinivasan

Department of Psychology

University of California, Berkeley

Berkeley, CA 94720-1650

Phone: 650-823-9488

Email: [email protected]


Abstract

Previous accounts of how humans develop expertise have focused on how deliberate

practice transforms the cognitive and perceptual representations and processes that give rise to

expertise. However, the likelihood of developing expertise with a particular tool may also

depend on the degree to which that tool fits pre-existing perceptual and cognitive abilities. The

present studies explored whether the abacus – a modern descendent of the first human computing

devices – evolved to exploit the constraints of human visual attention, or whether, instead,

abacus expertise involves extending or adapting the capacity of human visual attention through

practice. To address this question, we administered a series of visual search tasks to abacus

experts and subjects who had little to no abacus experience, in which search targets and

distractors were overlaid atop abacus “beads.” Across three studies, we found that experts and

naïve subjects were faster to detect targets in semantically-relevant regions of the abacus,

suggesting that the attentional biases that scaffold numerical processing in abacus experts require

little to no experience with the abacus to develop, and may emerge from general properties of

visual attention that are exploited by the design of the abacus itself.

Keywords: visual attention; mental abacus; expertise; artifact design; distributed cognition


One hallmark of human sophistication is our ability to create complex artifacts and to

develop expertise in using them. For example, auto mechanics are experts at using specialized

tools like car jacks and fender rollers, and neuroscientists are experts at using Magnetic

Resonance Imaging (MRI) and Transcranial Magnetic Stimulation (TMS). Previous studies of

expertise have explored how attaining expertise might depend on pre-existing individual

differences (e.g., Gobet & Ereku, 2007; Smith, Tsimpli & Ouhalla, 1993), but also on how it can

be developed through extensive, deliberate practice (Charness, Krampe & Mayr, 1996; Ericsson,

Krampe & Tesch-Romer, 1993; Ericsson & Lehman, 1996; Platz et al., 2014; Starkes et al.,

1996), and how such practice transforms the cognitive and perceptual representations and

processes that give rise to expertise (e.g., Chase & Simon, 1973; Chi, Glaser & Farr, 1988; Chi,

Glaser & Rees, 1982; Ericsson & Lehman, 1996; Gauthier, Skudlarski, Gore & Anderson, 2000).

However, the likelihood of developing expertise with a particular tool may also depend on the

degree to which that tool has been designed to fit the pre-existing perceptual and cognitive

abilities of novices. One tool that may exploit general properties of perception and cognition is

the abacus – a physical artifact that represents exact numerical quantities and arithmetic

operations over those quantities via the positions of beads in columns. The fact that the abacus is

a descendant of the oldest human computing devices (Ifrah, Harding, Bellos & Wood, 2000;

Menninger, 1969) raises the possibility that it may have evolved to become optimally tailored to

properties of human perception and cognition. To test this possibility, the present studies explore

the effects of artifact design and training on the development of expertise in mental abacus

(MA): A technique in which users visualize an abacus to perform mental arithmetic.

Understanding how MA expertise develops is important not only as a case study of

expertise, but also because it has been increasingly used as a math manipulative in educational


settings (see Ball, 1992; Uttal, Scudder, & Deloache, 1997), no doubt inspired by the remarkable

ability of MA experts to make calculations rapidly and accurately (Frank & Barner, 2011;

Hatano, Miyake & Binks, 1977; Stigler, 1984). However, it is not currently known what drives

MA expertise and thus how easily it can be attained in educational settings. If MA expertise

depends on perceptual and cognitive abilities that are unusual in the population or that require

extensive training, such expertise may be difficult to attain by ordinary children in standard math

classrooms. However, if the design of the abacus itself is well-suited to pre-existing perceptual

and cognitive abilities, MA expertise may be easier to attain, because it will be scaffolded by the

abacus structure.

In the present study, we addressed the effects of design and training on MA expertise by

exploring the role of visual attention in MA. MA places heavy demands on visual attention,

because encoding the numerical value of an abacus requires attending to a large number of

abacus beads. On the one hand, it is possible that through extensive training, expert MA users

have learned to attend to the abacus in particularly efficient ways, as has been documented in

other case studies of expertise. However, it is also possible that the design of the abacus itself

exploits basic attentional biases, leading even novices to attend toward semantically-relevant

aspects of the abacus, facilitating their understanding of how the abacus represents number. To

test these ideas, we explored how expert MA users and subjects with little to no experience with

the abacus attend to abacus-like displays, and thus whether MA practice shapes how attention is

allocated over the abacus.

The physical and mental abacus

Children learning MA are first taught how the physical abacus represents number, and are

instructed on how to perform basic arithmetic operations like addition and subtraction using


highly practiced physical procedures. The abacus represents numerical quantities via specific

arrangements of beads organized into columns (see Figure 1). Each column represents a digit

with a specific place value, which increases from right to left (i.e., from ones to tens to hundreds,

etc.). In the most commonly-used type of abacus, the Japanese Soroban, each column is divided

by a horizontal beam: below the beam are four earthly beads, and above the beam is one

heavenly bead. When the beads in each column are moved toward the horizontal beam to be in-

play, they count toward the value of the column, and can be used to represent a digit between 1

and 9. Specifically, when in-play, the heavenly bead in each column counts as a multiple of ‘5’

based on its place value (e.g., 5, 50, 500, etc.) and each of the four earthly beads counts as a

multiple of ‘1’ (e.g., 1, 10, 100, etc.). When all of the beads in a column are out-of-play – and

moved away from the horizontal beam – the column represents 0. Columns representing zero are

important for determining cardinality when the zero is trailing (e.g., the ‘0’ in 250) but not when

it is leading (e.g., 025).

Figure 1. A Soroban abacus with highlighted examples of in-play beads, out-of-play beads, and

beads in leading zero and trailing zero columns. This abacus represents the number “6,780”.

After MA students learn to use the physical abacus, they are trained on mental abacus.

0 6 7 8 0

leading zero column

trailing zero column

in-play

out-of-play


MA students learn to create a mental image of the abacus and to manipulate imagined beads to

perform mental arithmetic (Hatano et al., 1977). Experts are able to perform calculations with

staggering speed and accuracy (Frank & Barner, 2011; Hatano, Miyake & Binks, 1977; Stigler,

1984). For example, teenage MA users have placed first in the 2010 and 2014 Mental

Calculation World Cups, outpacing many older contestants.1 This expertise is impressive in part

because MA requires attending to, holding in memory, and updating the precise locations of a

large number of abacus beads in order to represent and manipulate exact numerical values. For

example, representing a cardinality like ‘699’ minimally requires representing the locations of 12

beads, and representing them as either in-play or out-of-play, heavenly or earthly, and having a

specific place value. The fact that MA users can mentally perform rapid and accurate

computations on such numbers – involving numerous abacus beads – is surprising in light of

previous studies, which have suggested that there are restrictions on visual attention and working

memory. Some studies, for example, indicate that humans can track only 3 or 4 objects

simultaneously (e.g., Cowan, 2001; Irwin, 1992; Luck & Vogel, 1997; Todd & Marois, 2004;

Vogel, Woodman & Luck, 2001; Vogel & Machizawa, 2004). Others indicate that although there

may not be a strict capacity limit on visual working memory, there is nonetheless a negative

relationship between the number of items we can store and the precision with which those items

are stored (see e.g., Alvarez & Cavanagh, 2004; Bays & Husain, 2008). Thus, by all previous

accounts, the ability of MA experts to mentally track the precise locations of numerous beads is

unexpected, raising questions about the mental structures and resources required to perform MA.

Although MA appears to exceed previously-reported restrictions on visual working

1 Mental Calculation World Cup – the World Championship for Mental Calculators. Retrieved

July 15, 2016, from http://www.recordholders.org/en/events/worldcup/index.html


memory, recent evidence suggests that it does rely primarily on visuo-spatial and motor

resources. First, several studies suggest that MA stands apart from other forms of mental

arithmetic, because it does not depend heavily upon the use of natural language. In particular,

while individuals who have not learned MA have great difficulty performing mental

computations under verbal interference, MA users are comparatively unaffected by verbal

interference tasks, but are instead more impaired by concurrent motor interference tasks (Hatano

& Osawa, 1983; Frank & Barner, 2011), which suggests that MA depends more on the motor

system than on natural language. Second, and consistent with the above, while standard methods

of arithmetic activate cortical areas associated with language and verbal working memory, a

large body of work suggests that MA selectively activates regions associated with visuo-spatial

and motor processes (Chen et al., 2006; Hanakawa et al., 2003; Hu et al., 2011; Ku et al., 2012;

Li et al., 2013a; Li et al., 2013b; Tanaka et al., 2002; Tanaka et al., 2012; Wu et al., 2009).

What is the relationship between MA training and visuo-spatial resources?

Given the fact that MA relies primarily on visuo-spatial resources and that such resources

in typical populations appear insufficient for tracking large numbers of abacus beads (Alvarez &

Cavanagh, 2004; Bays & Husain, 2008; Cowan, 2001; Irwin, 1992; Luck & Vogel, 1997; Todd

& Marois, 2004; Vogel, Woodman & Luck, 2001; Vogel & Machizawa, 2004), one possibility is

that attaining MA expertise requires unusually strong visuo-spatial abilities. For instance, those

who attain MA expertise might have particularly strong visuo-spatial abilities even before

learning MA, which would fit with other examples in which differences in attaining expertise

stem from pre-existing, individual differences (e.g., Gobet & Ereku, 2007; Smith, Tsimpli &

Ouhalla, 1993). Or, as has been documented in other case studies, MA expertise might require

extensive training (Charness, Krampe & Mayr, 1996; Ericsson, Krampe & Tesch-Romer, 1993;


Ericsson & Lehman, 1996; Platz et al., 2014; Starkes et al., 1996) that transforms cognitive and

perceptual representations and processes (Chase & Simon, 1973; Chi, Glaser & Farr, 1988; Chi,

Glaser & Rees, 1982; Ericsson & Lehman, 1996; Gauthier, Skudlarski, Gore & Anderson, 2000).

For example, one possibility is that MA learners develop augmented resources, including a more

precise ability to estimate numerosity, and expanded working memory capacities. Consistent

with this, some studies suggest that abacus training leads to structural changes in the brain,

including enhancements of the white matter tracts that link visuo-spatial and motor areas (Hu et

al., 2011; Li et al., 2013a; Li et al., 2013b). These cortical changes may lead to enhanced

resources, including more automatic numerical processing (e.g., of Arabic numerals; Wang et al.,

2013; Yao et al., 2015).

A third possibility, however, holds that MA training does not lead to augmented visuo-

spatial resources, and that MA experts need not be unusually gifted prior to first learning abacus.

Instead, MA may be designed to make efficient use of existing resources in typical populations

(Frank & Barner, 2011). For example, although MA experts process symbolic numerical stimuli

more automatically than naïve subjects who have had no experience using an abacus (Wang et

al., 2013; Yao et al., 2015), they are not faster or more accurate at estimating the cardinality of

sets of dots (Frank & Barner, 2011; Barner et al., 2016), suggesting that MA experts do not have

unusual perceptual expertise. Indeed, both naïve subjects and MA experts are better at numerical

estimation when dot arrays are configured similarly to the abacus – e.g., when dots are organized

into vertical columns – suggesting that the design of the abacus may be tailored to visuo-spatial

processing (Frank & Barner, 2011). Also consistent with the idea that MA training does not lead

to generally augmented visuo-spatial resources is evidence that children who are randomly-

assigned to learn MA do not develop greater numerical estimation abilities, visuo-spatial


working memory capacities, or mental rotation abilities than children who are taught arithmetic

using traditional methods (Barner et al., 2016).

Consistent with the idea that MA makes efficient use of existing resources, some

computational signatures of MA users adhere to previously-documented limits on visuo-spatial

working memory (Alvarez & Cavanagh, 2004; Bays & Husain, 2008; Cowan, 2001; Irwin, 1992;

Luck & Vogel, 1997; Todd & Marois, 2004; Vogel, Woodman & Luck, 2001; Vogel &

Machizawa, 2004). In particular, when performing arithmetic computations, MA users face

difficulty when addends contain more than 3 or 4 digits, i.e., where each addend is represented

by 3 or 4 abacus columns. This finding suggests that MA users may be able to track numerous

abacus beads (e.g., 12 in the case of ‘699’) because the design and structure of the abacus

encourages beads to be chunked into columns, such that each column can be treated a single

“object” in visuo-spatial working memory (Frank & Barner, 2011); By this account, MA users

may exhibit the limits of visual working memory when abacus columns are considered as units.

In the present study, we took a different approach toward understanding the effects of

design and training on MA expertise: Rather than testing whether MA training creates general

increases in visuo-spatial resources, we explored whether such training changes how visuo-

spatial resources are deployed when processing the abacus. Consistent with the idea that practice

using the abacus alters the ways in which MA users process the abacus, a recent study using a

Stroop-like paradigm found that when MA experts were shown an image of an abacus, they

immediately and involuntarily processed its represented number (Du, Yao, Zhang & Chen,

2014). In particular, compared to naïve subjects, MA experts were slower to decide which of two

abaci had a larger number of abacus beads when the abacus with the larger number of beads also

represented a smaller numeric value (compared to when it represented a larger numeric value; for


similar findings with Arabic numerals, see Henik & Tzelgov, 1982).

The finding described above suggests that practice using the abacus leads to automatic

processing of its numerical content. But does such practice change how MA users allocate visual

attention toward the abacus, or does the design of the abacus exploit basic attentional biases,

leading even novices to attend toward semantically-relevant aspects of the abacus? On the one

hand, abacus straining could affect how abacus users attend to the abacus: e.g., knowledge that

in-play beads (and not out-of-play beads) and beads in columns representing trailing zeroes (but

not leading zeroes) affect the number represented by the abacus could lead MA experts to

preferentially allocate their attention toward these aspects of abacus structure. This would be fit

with previous case studies showing that deliberate practice transforms domain-relevant

representations and processes (Chase & Simon, 1973; Chi, Glaser & Farr, 1988; Chi, Glaser &

Rees, 1982; Ericsson & Lehman, 1996; Gauthier, Skudlarski, Gore & Anderson, 2000),

including the allocation of attention task-relevant items (e.g., McCormack et al., 2014; Sheridan

& Reingold, 2007). Naïve subjects, who do not know how the abacus represents number, might

therefore fail to exhibit these attentional biases and might instead allocate their attention evenly

across the abacus. On the other hand, practice using the abacus need not shape low-level

attentional processes. Instead, the design of the abacus – having evolved over many centuries

(Ifrah, Harding, Bellos & Wood, 2000; Menninger, 1969) – may exploit attentional biases that

are shared by all humans. By harnessing such biases, the structure of the abacus could lead even

naïve subjects to preferentially attend to numerically relevant aspects of the abacus, like in-play

beads. These attentional biases could facilitate learning how the abacus represents number, and

with practice, processing of the abacus’ numerical content could become automatic.

What general aspects of visual perception and attention might lead even naïve subjects to


direct attention toward semantically-relevant regions of the abacus, like in-play beads? A large

literature on selective attention suggests that early visual processes create a saliency map of a

visual scene (Koch & Ullman, 1987; see also Itti & Koch, 2001; Itti et al., 1998; Li et al., 2002;

Treisman & Gelade, 1980), which identifies locations in a scene that are particularly conspicuous

relative to surrounding areas, and are thus more likely to attract attention – e.g., due to their

contrasting color, size, or orientation. In the context of the abacus, the horizontal beam appears

relatively conspicuous, given its distinct thickness and color, and contrasting orientation with the

vertical columns of beads (see Figure 1). Critically, if attention is allocated toward the horizontal

beam, other items in nearby locations might also be selected. This would predict an overall

attentional advantage for in-play beads, which are typically closer to the beam than out-of-play

beads. Moreover, because in-play beads within each column are directly adjoined to the

horizontal beam and to one another, they could together be treated as a single “object”. In light

of work suggesting that objects – and not merely spatial locations – can be the foci of attention

(for review, see Chen, 2012; Scholl, 2001; see also Duncan, 1984; Egly, Driver & Rafal, 1994;

Luck & Vogel, 1997; Treisman, 1982; Vecera & Farah, 1994), this would predict that attention

may automatically extend across all in-play beads.

The present studies

In the present studies, we set out to explore whether MA experts exhibit attentional biases

toward semantically-relevant aspects of the abacus, and if so, whether these biases emerge from

training using the abacus, or if they are instead shared by naïve subjects and exploited by the

design of the abacus itself. To measure how attention is allocated to the abacus, we conducted a

series of visual search tasks, and presented MA experts and naïve subjects with abacus-like

displays in which search targets and distractors were overlaid atop abacus “beads”. We reasoned


that participants would be faster to detect targets in locations toward which they preferentially

attend. Using this method, we asked several questions.

In Experiment 1, we asked how MA experts allocate their attention while reading the

abacus. To address this, MA experts were asked to complete a visual search task while

simultaneously reporting the number represented by the abacus. Experiment 2 then asked

whether the attentional biases exhibited by MA experts in Experiment 1 only emerge when

experts are explicitly asked to read the abacus, of if instead these biases may be more automatic.

To test this, MA experts were asked to complete only the visual search task, and were not asked

to report the number represented by the abacus. Finally, in Experiment 3 we explored whether

the ways in which experts allocate attention to the abacus are due to their experience using the

abacus, or if instead, the structure of the abacus itself mediates how visual attention is allocated.

To test this, we explored whether subjects with little to no experience or knowledge of the

abacus preferentially attend to semantically-relevant aspects of abacus structure, by having them

perform the visual search task from Experiments 1 and 2.

Experiment 1

The goal of Experiment 1 was to explore how abacus experts allocate their attention

toward the abacus when they are reading the number it represents, as an index of which aspects

of abacus structure they find most semantically-relevant. To test this, subjects were asked to

perform two concurrent tasks. First, they performed a visual conjunction search task (Treisman

& Gelade, 1980) in which search targets and distractors were overlaid atop schematic abacus

beads. We manipulated whether search targets were located on in-play or out-of-play beads, and

whether targets were located on beads in columns representing trailing or leading zeroes.

Second, on each trial of the visual search task, subjects were also asked to read and report the


number represented by the abacus, which changed from trial to trial. We reasoned that if in-play

beads and beads in trailing zero columns are especially relevant to extracting the number

represented by the abacus, then when experts are simultaneously reading the abacus, they should

preferentially attend to and thus be faster to detect targets at those locations, compared to when

targets are on out-of-play beads or on beads in leading zero columns.

Method

Participants. The participants were 63 MA students,2 who had a mean age of 10.62 years

(range: 8.42 to 15.0). All MA participants in this experiment and in Experiment 2 were enrolled

in Universal Computation Mental Arithmetic System (UCMAS) franchise schools, located in

Gujarat Province, India. MA students were included in our subject pool if they 1) had completed

Level 4 UCMAS training (which comprises training using the physical abacus, and an

introduction to MA), 2) were judged by their instructor to be among the best students in their

class, and 3) could travel to our test site in Vadodara, India. Seventeen of the participants from

Experiment 1 also participated in Experiment 2, with experiment order counterbalanced.3 Some

participants also received other measures, reported in Brooks et al. (under review).

Materials and Procedure. Participants completed two concurrent tasks. In the Visual

Search task, subjects were shown an abacus schematic in which targets and distractors were

overlaid atop abacus beads. Participants were asked to find the target item (i.e., a large circle)

and report whether it was red or blue, while ignoring distractor items (i.e., large squares and

small circles that could be either red or blue). Concurrently, in the Abacus Reading task, subjects

2 We lack demographic information for 10 of these participants 3 There were no main effects or interactions involving experiment order among the subjects who

participated in both Experiments 1 and 2.


were asked to report the number represented by the abacus schematic that was presented on each

trial.

Participants were seated in front of Macintosh laptop screens, wore headphones

throughout the task, and provided responses using the laptop keyboards and attached USB

numeric keypads. Stimuli were presented to subjects using custom software designed using the

Psychtoolbox 3 module of MATLAB (Kleiner et al., 2007). Prior to the task, small groups of

participants were given instructions by the experimenter in English (the language of instruction

at UCMAS), and these instructions were illustrated using examples of several trials of the Visual

Search and Abacus Reading tasks. Each participant then completed a brief training phase before

beginning the main task. During both training and test phases, feedback for a correct answer was

given with a green cartoon smiling face accompanied by a high pitch double tone, and feedback

for an incorrect answer was given with a red frowning face accompanied by a low pitch single

tone.

Training phase. The training phase included two stages. In the first stage, participants

received training on how to indicate the color of the target item during the Visual Search task.

Participants first received two trials in which they were shown either a blue or red large circle, in

succession, and were asked to press a button on the left or right side of their keyboard that

matched the circle in color (e.g., “Please press the “blue” button on the left”; the “blue” button

was the ‘z’ key, marked with a blue sticker, and the red button was the ‘/’ key, marked with a red

sticker). After this, participants received an additional six trials in which they were shown a large

circle and were asked to press the color-matching button (the circle was red on three trials and

blue on three trials). If a participant responded incorrectly on any trial, that trial was repeated.

Once all six trials were completed, participants proceeded to the second stage of training.


In the second stage of training, participants were introduced to the Visual Search and

Abacus Reading tasks. First, they were shown a schematic of an abacus, which served as the

search array. All of the abacus beads in the array except the target bead were overlaid with

distractor items. These distractors were randomly-generated, and could be either big squares or

small circles, and red or blue. The target bead was overlaid with a big circle, which was also

randomly assigned to be either red or blue. Participants were told that they would have to find

the big circle and then report the number of the abacus. The bead that contained the target item –

i.e., the big circle – was then identified with a white ring, and participants were asked to indicate

its color by pressing the appropriate “red” or “blue” button. After participants received feedback

for their response, the abacus search array was removed from the screen, and participants were

signaled to enter the number that had been depicted on the abacus, using their numeric keypad.

Participants received 10 additional trials of a similar structure, in which the abacus represented a

random 2 or 3 digit number, with the search target and distractors randomly distributed. For the

last eight of these trials, search targets were not identified for the participants with white rings,

but instead needed to be found. If participants responded incorrectly on any of the ten training

trials – e.g., by incorrectly indicating the target’s color or the abacus number – they began the ten

trial loop again. All participants successfully completed the training phase.

Test phase. On each of 128 trials, participants viewed either a two- or three-column

abacus schematic that depicted a number between 1 and 999 (see Figure 2).4 As in training,

participants were instructed to find the big circle and respond whether it was red or blue, while

ignoring the big square and small circle distractor items (which were randomly assigned to be

4 Single digit numbers (e.g., ‘1’, ‘9’, etc.) were represented with two abacus columns, using a

leading zero (e.g., ‘01’, ’09’).


blue or red). Reaction time and accuracy were recorded once the participant indicated the color

of the target; after this, the participant entered the number represented by the abacus and we

provided feedback and recorded their accuracy. As described below, we manipulated the location

of the target bead to explore how MA experts allocate visual attention between beads in leading

vs. trailing zero columns, and between in-play and out-of-play beads (see Figure 2).

Figure 2. Examples of trials from the search tasks of Experiments 1 through 3. Left Panel:

Example of a three-column trial in which the big circle (blue) is out-of-play and in a column

representing a trailing zero. Right Panel: Example of a two-column trial in which the big circle is

in-play.

Comparison of in-play vs out-of-play beads: In 64 of the trials, the target bead appeared

in a randomly-chosen column that represented a digit between 0 and 9. Across these trials, the

target bead appeared equally often in in-play vs. out-of-play positions, in two- vs. three-column

abacus schematics, and was also equally likely to appear in each of the rows of beads. Because


that abacus consists of one heavenly row of beads and four earthly rows of beads, this means that

the target was four times as likely to appear on an earthly bead as on a heavenly bead.

Comparison of beads in leading vs trailing zero columns: In the other 64 trials, the target

bead appeared in a column that represented either a leading or trailing zero, appearing in each of

these columns an equal number of times, and equally often in two- or three-column abacus

schematics. As in the in-play vs. out-of-play condition above, the target was also equally likely

to appear in each of the rows. Although the beads in both leading and trailing zero columns are

also necessarily out-of-play, we did not include trials in which targets appeared in these columns

in the in-play vs out-of-play analyses described below.

Results and Discussion

To begin, participants correctly reported the number represented in the abacus displays

on 95% of trials, indicating that they were engaged in reading these displays, as per our

instructions. Thus, we have reason to believe that during the search task, participants were

allocating their attention toward aspects of the abacus displays that are relevant to extracting

number.

Our analyses for the visual search task were restricted to trials in which participants

correctly indicated the color of the target, which occurred on 98% of trials. Reaction times were

log transformed prior to analyses. Following standard practice, in all experiments reported here,

all trials with log-transformed reaction times more than three standard deviations away from the

mean were excluded from analysis. In the final dataset, participants took on average 3.76

seconds (95% CI: 3.56 – 3.98) to indicate the color of the target. We used a linear mixed-effects

regression model (R version 3.3.0, lme4 version 1.1-12) to predict whether reaction times were

affected by the position of target beads: i.e., whether they appeared on in-play beads, out-of-play


beads, in leading zero columns, or in trailing zero columns, treating position on in-play beads as

the intercept. Additionally, our model tested for whether reaction times were affected by the trial

number of the task (to test for practice effects) and whether the abacus display had two or three

columns; we also tested for interactions between the number of abacus columns and position of

the target bead. We used the maximal convergent random effects structure (a random slope of

condition and a random intercept for each participant; following Barr et al., 2013). All p-values

were computed using the lmerTest package version . Raw data and full reports of analyses for all

experiments reported in this paper can be found at: https://github.com/langcog/abacus_attn.

Our analyses revealed that subjects were faster to detect targets on in-play beads

(M=3.63, 95% CI: 3.39-3.87) than on out-of-play beads (M=3.88, 3.67-4.12; β=.120, SE=.019,

p<.001; Figure 3), consistent with the idea that MA experts preferentially deploy their attention

toward in-play beads relative to out-of-play beads while reading the abacus. Further, while

participants’ speed to detect targets in trailing zero columns (M=3.66, 3.43-3.88) was similar to

their speed to detect in-play targets, they were significantly slower to detect targets on beads in

leading zero columns (M=3.86, 3.64-4.10; β=.154, SE=.018, p<.001; Figure 4), suggesting an

attentional advantage for beads in trailing zero columns relative to beads in leading zero

columns. Together, these results indicate that MA experts preferentially deploy their attention

toward in-play beads and beads in trailing zero columns to extract the number represented by the

abacus.

Also, our analyses indicated that participants were slower to detect targets when the

abacus display had three as opposed to two columns (β=.253, SE=.018, p<.001), which is

unsurprising given that the three-column displays had more distractors. Interestingly, however,

we also found significant interactions between number of abacus columns and target bead


position (all ps < .05). Specifically, the attentional advantage for in-play and trailing zero targets

were stronger in the two-column trials. We return to this finding when discussing the results of

Experiment 2.

Figure 3. Average reaction time to detect targets on beads that were in-play vs. out-of-play in

Experiments 1 and 2 (Error bars indicate 95% CI).

Figure 4. Average reaction time to detect targets on beads that were leading vs. trailing zero

columns in Experiments 1 and 2 (Error bars indicate 95% CI).

Experiment 1 (Dual task)

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Experiment 2 (Single task)

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Experiment 1 (Dual task) Experiment 2 (Single task)


Experiment 2

Experiment 1 suggested that, when asked to read the abacus, MA experts are faster to

detect search targets on in-play beads (than out-of-play beads) and on beads in columns

representing trailing zeroes (than on beads in columns representing leading zeroes), suggesting

that they preferentially allocate their attention toward these aspects of the abacus while

processing its number. In Experiment 2, we explored whether the biases observed toward in-play

beads and beads in trailing zero columns only emerge when experts are explicitly asked to read

the abacus, or if instead these biases may be more automatic and arise even when the intention is

not to read the abacus. As evidence for the latter possibility, recall that, in a stroop-like

paradigm, MA experts are slower to decide which of two abaci have a larger number of beads

when the abacus with the larger number of beads represents a smaller numerical value (Du et al.,

2014). This suggests that MA experts spontaneously read the number represented by an abacus

display, even when they are not instructed to do so. To test whether the attentional biases

observed in Experiment 1 might emerge automatically, in Experiment 2 we asked MA experts to

complete only the visual search task: i.e., we did not also ask them to report the number

represented by the abacus.

Method

Participants. The participants were 67 MA students, who had a mean age of 10.44 years

(range: 7.47 to 14.36). MA students were included in our subject pool as described in the

Participants section of Experiment 1.

Materials and Procedure. The methods and procedures were identical to those of

Experiment 1 except that participants were not also asked to report the number represented by

each abacus display, nor were they given training on this task.



As in Experiment 1, we restricted our analyses to trials in which participants correctly

indicated the color of the target, which occurred on 96% of trials. On average, in included trials,

participants took 1.97 seconds (1.85-2.09) to indicate the color of the target. As before, we used

a mixed-effects model to predict whether reaction times were affected by the position of target

beads (in-play, out-of-play, leading, or trailing), trial number, the number of abacus columns,

and interactions between number of abacus columns and target bead position. This model used

the same random effects structure as in Experiment 1: a random slope and intercept for each

participant.

Consistent with the findings of Experiment 1, we found that subjects were faster to detect

targets on in-play beads (M=1.88, 1.75-2.00) than on out-of-play beads (M=2.09, 1.97-2.21;

β=.119, SE=.020, p<.001; Figure 3). Further, while participants’ speed to detect targets in trailing

zero columns (M=1.89, 1.77-2.02) did not differ from their speed to detect in-play targets, they

were significantly slower to detect targets on beads in leading zero columns (M=2.05, 1.92-2.18;

β=.092, SE=.020, p<.001; Figure 4). Finally, as in Experiment 1, participants were slower to

detect targets in three-column than in two-column abacus displays (as in Experiment 1; β=.163,

SE=.019, p<.001); however, we did not find significant interactions between number of abacus

columns and target position (all ps>.54), indicating that attentional advantages for in-play and

trailing zero targets were equally strong across the two- and three-column trials. This contrasts

with Experiment 1, in which in-play and trailing-zero effects were weaker in three-column than

in two-column trials; This raises the possibility that the load imposed by having to read three-

column abaci in Experiment 1 may have suppressed attentional effects.


To directly assess the similarity of search patterns among experts who had been asked to

read the abacus (Experiment 1) and those who had not (Experiment 2) we used another mixed-

effects model to predict all participants’ reaction times from Experiments 1 and 2. This model

tested the effect of experiment, effects of target position, and interactions of experiment with

target position. Interestingly, although participants were unsurprisingly faster to detect the target

in Experiment 2 than in Experiment 1 (which required them to perform two concurrent tasks; β=-

.797; SE=.049; p<.001), the effects of target position did not interact with experiment (all

ps>.21), suggesting that subjects in the two experiments exhibited similar search patterns and

attentional biases. Taken together, these results suggest that, even when they are not explicitly

instructed to read the abacus, MA experts automatically deploy their attention toward

semantically-relevant aspects of abacus structure, like in-play beads and beads in trailing zero

columns.

Experiment 3

Experiment 3 explored how the automatic attentional biases documented in MA experts

in Experiments 1 and 2 relate to the extensive training those experts have received. On one hand,

practice using the abacus and knowledge of how the abacus represents number could create the

visual attentional biases observed in MA experts, consistent with previous case studies showing

that practice transforms the domain-relevant cognitive and perceptual processes that give rise to

expertise (Chase & Simon, 1973; Chi, Glaser & Farr, 1988; Chi, Glaser & Rees, 1982; Ericsson

& Lehman, 1996; Gauthier, Skudlarski, Gore & Anderson, 2000; McCormack et al., 2014;

Sheridan & Reingold, 2007). By this account, extensive training could lead MA users to

preferentially allocate their attention toward semantically-relevant aspects of abacus structure,

like in-play beads, and beads in columns representing trailing zeroes. This account would then


predict that naïve participants who do not know how the abacus represents number should fail to

exhibit equivalent attentional biases, and might instead allocate their attention evenly across the

abacus. On the other hand, abacus training may not shape visual attentional processes, but might

instead be scaffolded from basic properties of visual attention. In particular, the biases observed

in experts may not require extensive practice using the abacus, but may instead be present in

experts and non-experts alike, and stem from the design and structure of the abacus itself. If so,

then even naïve subjects may preferentially attend to semantically-relevant aspects of the abacus,

facilitating the efforts of naïve users to learn MA.

To explore these questions, we conducted the visual search task of Experiments 1 and 2

with a group of adult participants from an American university community, and probed their

familiarity with the abacus after they performed the task. This method allowed us to address

whether biases toward in-play beads and beads in trailing zero columns are present in naïve

subjects with no familiarity with the abacus, and if not, what levels of familiarity with the abacus

might be required to develop such biases.

Method

Participants. Fifty-six adult participants (mean age = 21.24; range 18.42 – 33.27) were

recruited from a participant database at the University of California, San Diego. All participants

received course credit for their participation.

Materials and Procedure. All participants completed the visual search task prior to

answering questions about their familiarity with the abacus.

Visual Search Task. The materials and procedures for the visual search task were the

same as in Experiments 1 and 2.


Abacus Familiarity Assessment. After they completed the visual search task, we asked

participants two questions to determine their level of familiarity with the abacus. First, we

probed whether participants recognized that the display from the Visual Search Task represented

an abacus: Do you know what the display was? Next, after telling them that the display depicted

an abacus (if they had not said so themselves), we asked participants to describe their experience

using an abacus: Have you ever used an abacus, and if so, for how long?

Based on their answers to these questions, we classified participants into three categories:

“naïve”, “low familiarity” and “moderate familiarity”. Participants were considered “naïve” with

respect to the abacus if they answered no to both questions (n = 32). They were considered to

have “low familiarity” with the abacus (n = 14) if they stated that the display was an abacus (in

addition to “abacus”, we accepted responses such as “counting beads” and “old calculator”) or

reported using an abacus in the past despite failing to initially recognize that the display was an

abacus. Finally, participants were considered to have “moderate familiarity” with the abacus (n =

10) if they both recognized that the display was an abacus and reported having used an abacus in

the past. In general, even participants in the “moderate familiarity” group reported only sparse

levels of prior experience with the abacus: e.g., four out of the seven participants in this group

reported using the abacus only briefly as young children.


As before, we restricted our analyses to trials in which participants correctly indicated the

color of the target, which occurred on 98% of trials. On included trials, participants took on

average 1.20 seconds (1.12-1.27) to indicate the color of the target. We again used a mixed-

effects model with the same random effects structure to predict whether reaction times were

affected by the position of target beads (in-play, out-of-play, leading, or trailing), trial number,


familiarity with the abacus (naïve, low, moderate), and by interactions between abacus

familiarity and target bead position.5

Just as in Experiments 1 and 2 with MA experts, participants were faster to detect targets

on in-play beads (M=1.15, 1.07-1.25) than on out-of-play beads (M=1.23, 1.15-1.30; β=.077,

SE=.017, p<.001; Figure 5). Also as before, while participants’ speed to detect targets in trailing

zero columns (M=1.16, 1.10-1.24) did not differ from their speed to detect in-play targets, they

were significantly slower to detect targets on beads in leading zero columns (M=1.23, 1.16-1.32;

β=.056, SE=.017, p<.005; Figure 6).

Interestingly, naïve subjects (who have never used an abacus and cannot recognize one)

showed just as strong of an advantage for in-play beads as did subjects in the low and moderate

familiarity groups (Figure 5): There were no significant interactions between levels of abacus

familiarity and position on out-of-play (as opposed to in-play) beads (all ps > .37). In contrast,

there was a significant interaction between abacus familiarity and target position in leading vs.

trailing columns, as subjects in the moderate familiarity group exhibited a greater trailing vs.

leading effect, compared to naive subjects (β=.116, SE=.044, p=.010). To test whether even the

naïve subjects (N=32) showed the leading zero effect, we fit a follow-up model to the data from

only these participants and again found a significant leading zero effect (β=.056, SE=.015,

p<.001; Figure 6).

5 A preliminary model that included number of abacus columns did not yield any significant

interactions with effects of target position (as in Experiment 2); This factor was excluded from

subsequent models.


Figure 5. Average reaction time to detect targets on beads that were in-play vs. out-of-play by

subjects in the naïve, low familiarity, and moderate familiarity groups of Experiment 3 (Error

bars indicate 95% CI).

Figure 6. Average reaction time to detect targets on beads in leading vs. trailing zero columns by

subjects in the naïve, low familiarity, and moderate familiarity groups of Experiment 3 (Error

bars indicate 95% CI).

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Naive Low Familiarity Moderate Familiarity

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To directly assess the similarity of search patterns between the subjects of Experiment 3 –

who had little to no experience using the abacus – and the MA experts from Experiment 2, we

used a final mixed-effects model. This model tested the effect of experiment, effects of target

position, and interactions of experiment with target position. Although participants from

Experiment 3 were on average faster to detect the target than MA experts in Experiment 2 (β=-

.450; SE=.047; p<.001),6 the effects of target position did not interact with experiment (all

ps>.05), suggesting that subjects in the two experiments exhibited similar search patterns and

attentional biases. Taken together, these results suggest that the automatic attentional biases that

scaffold numerical processing in abacus experts are present in naïve subjects.

Post-hoc Analyses of the In-play Bias

Thus far, we have explored how MA experts (Experiments 1 and 2) and participants with

little to no familiarity with the abacus (Experiment 3) allocate their attention toward the abacus.

Our evidence suggests that a bias to attend toward in-play beads relative to out-of-play beads is

present in entirely naïve participants, as is a bias to attend toward trailing zeroes relative to

leading zeroes (though only the former is equally strong among entirely naïve subjects and

subjects with more familiarity with the abacus; Experiment 3). This pattern of findings suggests

that in some cases, the design of the abacus takes advantage of general attentional biases we all

share to direct attention toward semantically-relevant aspects of abacus structure. Here, in a

series of post-hoc analyses, we explored what more general attentional biases might underlie the

advantage for in-play beads, and if the same biases guide both expert MA users and participants

with little to no experience with the abacus. 6 This finding could indicate that MA experts – but not subjects with little to no familiarity with the abacus – automatically process the abacus prior to searching for the target. However, the slower speed of MA experts in Experiment 2 could also be explained by the fact that those subjects were children, while the participants in Experiment 3 were adults.


To begin, we explored whether the advantage for in-play beads could be explained by a

general attentional bias that favors targets that are closer to the horizontal beam over those that

are further away. In particular, attention may be drawn first to the horizontal beam due to its

relative salience (e.g., it has a distinct thickness and color, and contrasts in orientation with

vertical columns of beads; see Koch & Ullman, 1987; see also Itti & Koch, 2001; Itti et al., 1998;

Li et al., 2002; Treisman & Gelade, 1980), such that locations further from the beam are less

selected for attention than locations closer to the beam. This would explain the in-play

advantage, because in-play beads are generally closer to the horizontal beam than out-of-play

beads. But critically, if this is correct, then in some cases out-of-play targets should be detected

faster than in-play targets due to the in-play beads being further away from the beam, i.e., in

cases where several earthly beads within a column are in-play, and the target in-play bead does

not directly adjoin the beam.

To test the prediction that proximity to the beam explains the in-play advantage over and

above in-play/out-of-play status, we used mixed-effect models to explore whether subjects’

speed to detect the target in each experiment was affected by in-play/out-of-play status (we

excluded leading / trailing trials from analysis), trial number, proximity of the target to the beam,

and an interaction between in-play/out-of-play status and proximity to the beam. This analysis

was confined to earthly beads, since in-play status and proximity to the beam are confounded for

heavenly beads. Consistent with the idea that the in-play advantage can be explained by a bias to

attend toward targets that are closer to the horizontal beam, our models for Experiments 1 and 3

detected effects of proximity to the beam (Experiment 1: β=.030, SE=.011, p=.005; Experiment

3: β=.044, SE=.009, p<.001) but no effect of in-play/out-of-play status (Experiment 1: β=-.005,

SE=.077, p=.94; Experiment 3: β=.090, SE=.069, p=.19), or interactions between in-play/out-of-


play status and proximity to the beam (Experiment 1: β=.005, SE=.015, p=.75; Experiment 3:

β=.009, SE=.013, p=.48). These findings suggest that, for the MA experts in Experiment 1 (who

performed the dual search-reading task), and the naïve subjects in Experiment 3 (who performed

only the search task), beads that were closer to the beam were more likely to be selected for

attention, suggesting that the in-play advantage emerges from this general attentional bias.

Interestingly, however, our model of the search patterns of MA experts in Experiment 2

(who performed only the search task, just like the mostly naïve subjects of Experiment 3

described above) yielded a different pattern of effects. Here, proximity to the beam did not

emerge as a significant predictor (β=.004, SE=.011, p=.71), though in-play/out-of-play status did

(β=-.279, SE=.084, p=.001). The in-play effect was also qualified by an interaction between in-

play/out-of-play status and proximity to the beam (β=.059, SE=.016, p<.001), such that

participants were slower to detect out-of-play targets – but not in-play targets – that were further

away from the beam. The fact that proximity to the beam did not affect MA experts’ speed to

detect targets on in-play beads in Experiment 2 suggests that they automatically allocated their

attention across these beads, and perhaps treated them as a “single object” of attention (for

review, see Chen, 2012; Scholl, 2001; see also Duncan, 1984; Egly, Driver & Rafal, 1994; Luck

& Vogel, 1997; Treisman, 1982; Vecera & Farah, 1994).

In sum, our post-hoc analyses provide some evidence that the in-play advantage observed

in MA experts in Experiment 1 and entirely naïve subjects in Experiment 3 might be explained

by a more general attentional bias to attend toward salient objects (e.g., the horizontal cross-

beam), such that other proximal objects are also likely to be selected for attention (e.g., in-play or

out-of-play beads that are close to the beam), since effects of proximity to the beam emerged

over and above in-play/out-of-play status in both experiments. However, the fact that MA


experts from Experiment 2 – who performed an equivalent task to the naïve subjects from

Experiment 3 – did not show an effect of proximity to the beam on in-play targets suggests

another possibility: As a consequence of training, MA experts may automatically allocate their

attention across in-play beads, by selecting these beads as a single “object” of attention.

General Discussion

The present studies explored whether the abacus – a modern descendent of the first

human computing devices – evolved to exploit the constraints of human visual attention, or

whether, instead, abacus expertise involves extending or adapting the capacity of human visual

attention through practice. To address this question, we administered a series of visual search

tasks to MA experts and subjects who had little to no experience with the abacus. In these tasks,

search targets and distractors were overlaid atop abacus “beads.” Using this method, Experiment

1 found that when asked to read an abacus, MA experts were faster to detect targets that were in-

play, and targets in columns representing trailing zeroes, providing evidence that these aspects of

abacus structure are relevant to extracting abacus number. Experiment 2 built upon these results

and found that, for MA experts, attentional biases toward in-play and trailing zero beads may be

automatic, as they emerge even when experts are not explicitly asked to read the abacus. Finally,

in Experiment 3 we found that, like MA experts, subjects with no experience or familiarity with

the abacus also show an advantage for in-play beads, and for beads in trailing zero columns

(though the latter tendency is stronger when subjects have more familiarity with the abacus).

Together, these findings suggest that the automatic biases that scaffold numerical processing in

MA experts require little to no experience with the abacus to develop, and thus emerge from

general properties of visual attention that are exploited by the design of the abacus itself.


By suggesting that the development of MA expertise is in part facilitated by the design of

the abacus, our results complement previous accounts of the development of expertise, which

have argued that expertise may stem from individual differences (Gobet & Ereku, 2007; Smith,

Tsimpli & Ouhalla, 1993), or require extensive, deliberate practice (Charness, Krampe & Mayr,

1996; Ericsson, Krampe & Tesch-Romer, 1993; Ericsson & Lehman, 1996; Platz et al., 2014;

Starkes et al., 1996) to transform domain-relevant representations and processes (Chase &

Simon, 1973; Chi, Glaser & Farr, 1988; Chi, Glaser & Rees, 1982; Ericsson & Lehman, 1996;

Gauthier, Skudlarski, Gore & Anderson, 2000; McCormack, 2014; Sheridan & Rheingold,

2014). Consequently, our results have implications for how easily and widely MA expertise can

be attained, and thus its educational utility. In particular, by encouraging even novices to

preferentially attend to numerically-relevant aspects of the abacus, like in-play beads and beads

in trailing zero columns (Experiment 3), the design of the abacus may scaffold learning how the

abacus represents number, and thus make MA expertise more easily attainable. This feature of

the abacus may hold broader lessons for the use of concrete manipulatives in mathematical

education (Ball, 1992; Uttal, Scudder, & Deloache, 1997), as it suggests that such manipulatives

will be most effective when their design takes into account the perceptual and cognitive abilities

of their users.

What general aspects of visual attention might give rise to the advantage for in-play

beads observed in both MA experts and naïve subjects? Our post-hoc analyses suggest that one

critical factor in explaining the in-play advantage is the fact that beads that are in-play tend to be

closer to the horizontal beam than beads that are out-of-play. Indeed, we found that when

proximity to the beam was entered in as a predictor of reaction times, the in-play advantage

vanished for MA experts in Experiment 1 and naïve subjects in Experiment 3, suggesting that the


in-play advantage observed in these experiments reduced to a bias to attend toward locations that

are closer to the horizontal beam. To explain this, we have suggested that attention may be

initially drawn to the beam because of its unique thickness, color, and orientation – e.g., early

visual processes could identify the beam as the most salient region of the visual scene (see Koch

& Ullman, 1987; see also Itti & Koch, 2001; Itti et al., 1998; Li et al., 2002; Treisman & Gelade,

1980) – making other neighboring areas, like in-play beads, more likely to be selected for

attention. One intriguing possibility left open by our data is whether attention extends

automatically across all in-play beads for MA experts, because the beam and in-play beads are

treated as a single “object” of attention (for review, see Chen, 2012; Scholl, 2001; see also

Duncan, 1984; Egly, Driver & Rafal, 1994; Luck & Vogel, 1997; Treisman, 1982; Vecera &

Farah, 1994).

Regardless of what biases in visual attention give rise to the in-play advantage, our

results suggest that the design of the abacus capitalizes on such biases to direct attention toward

semantically-relevant aspects of the abacus, like in-play beads. Is this simply a coincidence? One

intriguing possibility is that the structure of the abacus evolved over time, and became optimally

tailored to properties of visual attention and human cognition more generally (Frank & Barner,

2011). By this account, early versions of the abacus may not have been as easy to use or as

powerful as current forms (see Ifrah, Harding, Bellos & Wood, 2000), and these perceived

shortcomings may have driven further innovation and changes to the basic design of the abacus,

as appears to be the case with the design of many artifacts (Basalla, 1988; Petroski, 1993).

Consistent with the idea that the design of the abacus could have evolved over time, the

abacus has a long cultural history (Ifrah, Harding, Bellos & Wood, 2000; Menninger, 1969), as

early variants were present in ancient Mesopotamia, Greece (the Salamis Tablet), Rome (the


Roman Hand Abacus), and China (the Suan Pan). Further, it is known that current versions of

the abacus, including the specific abacus studied here – the Japanese Soroban – have been

modified from their early forms, e.g., by reducing the number of beads in each column (Frederic,

2005). Such modifications may have resulted in design features that have made the abacus

optimally tailored to properties of visual attention and human cognition more generally. For

example, both the Suan Pan and the Soroban include a salient horizontal divider and vertically-

oriented columns of beads, and these features could mediate how visual attention is allocated:

The salience of the horizontal divider could attract attention to in-play beads, and the vertical

grouping of columns of beads could encourage columns to be treated as “objects” in working

memory. Although little is currently known about the degree to which psychological factors have

shaped historical modifications in abacus design across cultures, it would seem logical to think

that they have, since computations using the physical abacus are distributed (Hutchins, 1995;

Zhang & Norman, 1994), and require coordination among the internal mental representations of

abacus users and the external knowledge representations instantiated by the abacus.

More centrally, another reason to think that the abacus may have evolved to coordinate

well with human perception and cognition is the fact that MA is possible at all – i.e., that

individuals can perform computations using mental images of an abacus – and that MA expertise

can be widely attained. If the abacus was difficult to hold in memory and process, MA would be

difficult, and MA expertise might only be attained by those with unusually strong visuo-spatial

resources. However, as reviewed in the Introduction, acquiring MA expertise does not require

unusual visuo-spatial resources, such as augmented spatial working memory or mental rotation

capacities; Instead, even children with typical levels of visuo-spatial ability can readily acquire

MA expertise (Barner et al., 2016). Further, MA experts do not exhibit unusual perceptual


expertise, but instead resemble naïve subjects. MA users are not faster or more accurate at

estimating the cardinality of dot arrays than naïve subjects (Frank & Barner, 2011); Instead, both

naïve subjects and MA experts are better at numerical estimation when dot arrays are configured

similarly to the abacus (Frank & Barner, 2011). In sum, by showing that the attentional biases

exhibited by MA experts are shared by novices, the present studies complement previous

findings, and suggest that the abacus may have evolved to make optimal use of pre-existing

visuo-spatial resources.


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